Given that the length of the cube's edges is 
, then the length of the diagonal along any face of the cube is, by the Pythagorean theorem,
Then the diagonal along the opposite vertices of the cube (i.e. the line segment through the center of the cube joining its opposite corners) is, again by the Pythagorean theorem,
The radius of the sphere,
, corresponds to half the length of the cube's diagonal; that is,
The surface area of the cube in terms of its edge length is
which means the area, rewritten in terms of the sphere's radius, is
Hence the rate of change of the cube's surface area with respect to a change in the sphere's radius is
When
, we have
while when
, we get
I'm not sure what part (d) reads, so I'll leave that to you...
Then the diagonal along the opposite vertices of the cube (i.e. the line segment through the center of the cube joining its opposite corners) is, again by the Pythagorean theorem,
The radius of the sphere,
The surface area of the cube in terms of its edge length
which means the area, rewritten in terms of the sphere's radius
Hence the rate of change of the cube's surface area with respect to a change in the sphere's radius is
When
while when
I'm not sure what part (d) reads, so I'll leave that to you...
